Anton Fernandez

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```					 Effects and correction of
sampling error in individual
tree mortality models
Clara Antón Fernández
Robert E. Froese
School of Forest Resources and Environmental Science.
Michigan Technological University
1400 Townsend Drive. Houghton, Michigan 49931

Western mensurationist
June 25, 2007
The problem
The problem
• Competition
variables
are
measured
with error
The problem
•   Min 45
•   Max 109
•   Mean 70
•   Measured 65.4
The problem
• Prediction of a response versus inference for
parameters
– Generally, there is no need for the modeling of
measurement error to play a role in the prediction
problem
– The unique situation when we need to correctly model
the measurement error occurs when we develop a
prediction model using data from one population but
we wish to predict in another population.
The problem
• Sampling error variances change during
the simulation.
• They depend on
– sample plot sizes (fixed at the beginning of
the simulation but may be differ from the one
used for fitting the model)
– spatial structure of the stand (tree size and
spacing)
The cost
The cost
• If we ignore the changes in the error
structure of the competition variables

DURING MODEL                    LOSS OF
FITTING                         POWER
for detecting relationships
among variables

PREDICTION                         BIASED
The cost

TRUE

OBSERVED

Source: Carroll, R. J., D. Ruppert, L. A. Stefanski, and C. M. Crainiceanu. 2006.
Measurement error in nonlinear models. Chapman and Hall/CRC.
The linear case
diameter increment model
Solutions: Linear case
• Attenuation: The effect of measurement
error is, generally, to bias the slope
estimate towards zero.
• Stage and Wykoff (1998) proposed the
Structural Based Prediction (SBP)
procedure
• Results: considerable change in the
magnitude of some regression coefficients
and an increase in residual variance
Effects of measurement error

The effects of measurement
error are complex

– The bias could be under or over-
estimated, even for the variables
that are measured without error.
The non-linear case
mortality model
The non-linear case
• Regression Calibration                 Simple
Once the replacement is      Generally applicable
made, essentially the same
• SIMEX     methods for ongoing
Computationally
analyses can be employed
more intensive that RC
as if X was observed

• Score function methods                      Result in fully
consistent estimators
more generally
• Likelihood and quasilikelihood                Computationally
more demanding

Require strong
• Bayesian methods                              distributional
assumptions
Regression Calibration
• “Widely used, effective (and) reasonably
well investigated” (Pierce and Kellerer, 2004)
• Basis: replacement of X by the regression
of X on (Z,W). X variables measured with error, Z variables
measured without error, W observation related with X

• Once the replacement is made, essentially
the same methods for ongoing analyses
can be employed as if X was observed.
Results
Data
• USDA Forest Service Region 1
Permanent Plot Program
• The set includes
– regenerating stands in the Rocky Mountain
Region
– control and treated (managed) stands
• 34,243 tree measurements
• 189 stands
Results and consequences
BEFORE                                                   AFTER RC
400

400
300

300
Frequency

Frequency
200

200
100

100
0

0    20     40          60   80   100               0     0   20     40          60   80   100

PBAL                                                     PBAL

PBAL frequency distribution for western hemlock
Results and consequences
Western hemlock
Results and consequences
Lodgepole pine
Results and consequences
• The effect of the sampling error in the
multivariate logistic case can be under- or
overestimate the effect of the variable,
even for variables that are measured
without errors
• Results might be influenced by the limited
scope of the data
Summary
Measurement error in mortality models
• Measurement error can cause
– Loss of power in the fitting phase
– Bias in the prediction phase
• Regression Calibration corrects for measurement error
before models are fitted or applied
• The effect of the sampling error in the multivariate
logistic case can be under- or overestimate the effect of
the variable, even for variables that are measured
without errors

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